Wahl maps and extensions of canonical curves and K3 surfaces

Abstract

Let $C$ be a smooth projective curve of genus $g\ge 11$, non-tetragonal, considered in its canonical embedding in ${\mathbf{P}}^{g-1}$. We prove that $C$ is a linear section of an arithmetically Gorenstein normal variety $Y$ in ${\mathbf{P}}^{g+r}$, not a cone, with $\dim (Y)=r+2$ and ${\omega }_Y={\mathcal{O}}_Y(-r)$, if the Gauss--Wahl map of $C$ has corank larger or equal than $r+1$. This relies on previous work of Wahl and Arbarello-Bruno-Sernesi; a partial converse is given via a theorem of Lvovski. We derive a similar result for $K3$ surfaces: Let $(S,L)$ be a polarized $K3$ surface of genus $g\ge 11$, non-tetragonal, and considered in its embedding in $|L{|}^{\vee }\cong {\mathbf{P}}^g$. It is a linear section of a variety $Y$ as above if $H^1(T_S\otimes L^{\vee })$ has dimension larger or equal than $r$. We give various applications, including one to the following forgetful modular map: Let ${\mathcal{K}}_g$ be the moduli space of polarized $K3$ surfaces of genus $g$, and ${\mathcal{K}\mathcal{C}}_g$ the space of pairs $(S,C)$ with $C$ a smooth curve on $S$ and $(S,{\mathcal{O}}_S(C))\in {\mathcal{K}}_g$; we consider the map $c_g:(S,C)\in {\mathcal{K}}_g\mapsto C\in {\mathcal{M}}_g$. If $g\ge 11$, we show that this map has smooth fibres over the locus of non-tetragonal curves, with fibre-dimension over non-tetragonal $C$ the corank of the Gauss--Wahl map of $C$ minus one.

v3: consequent pruning in the presentation following journal's recommendation, substance unchanged; v4: post-final version, comments on Prop.2.4 and additional details in the proof of Prop.8.4 added

Cited by in corpus (3)

• $K3$ curves with index $k>1$
• Maximal variation of curves on K3 surfaces
• Deformations and extensions of Gorenstein weighted projective spaces